Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×#120 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1031 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,12)(8,13)(9,17)(10,14)(11,15)(16,19)(18,20);(5,7)(6,11)(8,15)(9,18)(10,16)(12,17)(13,19)(14,20);(2,4)(5,7)(6,12)(8,15)(9,18)(10,16)(11,13)(14,19)(17,20);(5,8,10,9)(6,13,14,17)(7,15,16,18)(11,19,20,12)Polycyclic presentation:
2^8|d^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gh,c^b=cg,e^b=eh,e^c=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #359067, #359068, #375819, … |
| maximal normal subgroup of | #359067, #359068, #375819, … |
| maximal quotient of | #233630, #233634, #233637, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16387);